Your grammar book will have several pages on this topic. Briefly, the simple present and present continous can be used for "scheduled" future events (which are viewed as certain to occur). The simple present usually applies to transportation and the present continuous usually applies to people. Examples: "My plane leaves at 0920 on January 3, 2019." (several months from now, but I have a ticket, so it is scheduled future event) "My wife and I are leaving on January 3, 2019 and we are returning on January 17." The near future (going to <verb>) can be used for planned near future and far future events (which are viewed as certain or probable). Examples: "After supper, I am going to read a book." "My sister is going to marry her boyfriend in January." The near future can also be predictions about the near future. Example: "The clouds are dark. It is going to rain soon." The simple future (will <verb>) can be used for intended future events (which are view as general intentions without a plan). Examples: "My wife will visit her sister next summer." (A general intention). There are additional uses of the simple future such as to make a decision or to make a promise. For more information, consult a grammar book.

Yes, you can certainly say "I'm having a test tomorrow". It's quite normal to use the present continuous for a future arrangement.

US - You can't say "I'm having a test tomorrow." That is not a standard American English construction. "I have a test tomorrow." or "I will have a test tomorrow." are ok. Contracting "I will" to "I'll" would be a little strange in this sentence, but you could do it. If you said "I'm having a test tomorrow." Someone might reply "I hope it isn't in grammar, because you need to do some more studying."

No, no. Its about NOW, not tomorrow. I have test tomorrow.